Lecce winger Remi Oudin cancelled out a goal by striker Ciro Immobile before halftime, adding another one shortly after the break before Milinkovic-Savic levelled the score again in stoppage-time.
Lazio, who have earned only four points in their last five games, remain third on 65 points, two ahead of fourth-placed Inter Milan who host Sassuolo tomorrow.
“I’m not so angry because in the end we won a point that can come in handy,” Lazio coach Maurizio Sarri told DAZN.
“The negative aspect is the naivety with which we took the goal for 1-1.
“If we had equalised five minutes earlier we would have won, that was the feeling.”
The visitors could have taken a surprise lead in a mostly dull first half when the referee pointed at the penalty spot for a foul on Alexis Blin but Gabriel Strefezza, Lecce’s top scorer with eight goals this season, sent the ball wide of the left post.
Instead, Immobile put Lazio in front in the 34th minute from a neat Luis Alberto cross inside the box.
Immobile, who also netted an opener in the reverse fixture, scored his 11th goal this season and his first in a home league match since September, prompting the fans to chant his name.
The home crowd’s joy was short-lived though as Lecce equalised in first half stoppage-time through winger Oudin, who surprised Lazio keeper Ivan Provedel with a low shot from the edge of the box.
Lecce continued to push and Oudin put them ahead six minutes into the second half with a single-touch goal.
Lazio, who kept between 60% and 70% of possession throughout the game, repeatedly struggled in front of goal and forward Pedro hit the post from a distance in stoppage-time.
But Milinkovic-Savic salvaged one point for the home side just before the final whistle, netting on a rebound with a header.
“I would have liked to make it 3-2,” he said.
“We conceded two rather silly goals, the second after our own mistake. We need to improve our concentration and try to always bring home the three points.”
Lazio next travel to Udinese on May 21 with three games remaining in the season.
Lecce, who are five points above the drop zone, host 18th-placed Spezia on the same day.